Question

What is the resultant of two vectors $\vec A$ and $\vec B$, when they are in
(a) Same direction
(b) Opposite direction

Answer 1

Hint: Vectors are quantities that correspond to both magnitude and direction. The properties of vectors and the concept of resultant vector of two or more vectors acting from the same location is applied in order to determine the resultant.

Complete step by step answer:
The above problem revolves around the concept of vectors and their corresponding resultant if there are more than two vectors that are present. In order to find the resultant of the two vectors $\vec A$ and $\vec B$, we first need to know what a resultant vector is.

Resultant vectors are vectors that give the total magnitude of a number of vectors connected together. It is the result vector that is obtained by summing up the magnitudes of two or more vectors together. Here, we are asked to determine the resultant, that is, the total magnitude due to vectors, $\vec A$ and $\vec B$ as well as the direction in which the resultant vector is pointed toward.

The resultant vector would be the total vector that is deduced from dissolving all the vectors connected together into one single vector with some magnitude and direction. Vectors are basically pointers that specify a certain distance or in other words a certain magnitude value and also shows the direction of travel of the vector quantity.

We are asked to consider two vectors in this case that is vector $\vec A$ and vector $\vec B$ and based on the direction of these vectors we need to determine the resultant vector that will be a single vector which will represent these two vectors by a single magnitude value and direction.

(a) Here, the vectors $\vec A$ and $\vec B$ are given to be in the same direction. When two vectors are said to be in the same direction then they are said to be added up together to give the resultant vector. The magnitude of the will be the sum value after adding the magnitudes of the two vectors respectively. Thus:
$\left| {\vec R} \right| = \left| {\vec A} \right| + \left| {\vec B} \right|$
Where, $\vec R$ denotes the resultant and $\left| {\vec R} \right|$ denotes the magnitude of the resultant vector.

Now, we come to the direction of the resultant vector. Since the given vectors are said to be in the same direction the resultant vector will also be in the same direction but the magnitude will be the summation. Hence, for vectors $\vec A$ and $\vec B$ the resultant vector $\vec R$ will have a magnitude $\vec A + \vec B$ in the same direction as either $\vec A$ or $\vec B$ since both directions are the same.

(b) Here, the vectors $\vec A$ and $\vec B$ are given to be in opposite directions to each other. This means that the resultant of the two vectors will be given by the difference between the two vectors since they are in two different exactly opposite directions. The magnitude of the resultant is thus given by the difference of the vectors $\vec A$ and $\vec B$.

However, there will be a need to determine which vector has a greater magnitude in order to know which vector must be subtracted from which. Hence, we encounter two cases over here. One case is when the vector $\vec A$ has a greater magnitude than vector $\vec B$ or vice versa. Thus, we have:
When, $\left| {\vec A} \right| > \left| {\vec B} \right|$ then the resultant vector $\vec R$ is given by:
$\left| {\vec R} \right| = \left| {\vec A} \right| - \left| {\vec B} \right|$
Similarly, when $\left| {\vec B} \right| > \left| {\vec A} \right|$ then the resultant vector $\vec R$ is given by:
$\left| {\vec R} \right| = \left| {\vec B} \right| - \left| {\vec A} \right|$

Thus, we can see that when the vector with the lesser magnitude compared to the other is subtracted from the vector with more magnitude the resultant magnitude is obtained.Hence, the magnitude of the resultant vector will be the difference between one another.Now, coming to the direction, the direction of the resultant vector is always in the direction of the vector with the greater magnitude relative to the other vector. This will be the resultant direction.

Note: A point to note is that when the vectors are said to have the same direction but are equal in magnitude then they cancel out each other and a null vector is obtained which signifies that the value of the resultant vector is zero. Vectors are highly useful tools to represent real life vector quantities which have a certain value and direction, that is, for example velocity, displacement, linear momentum etc.